3.3.43 \(\int \frac {1}{\sqrt {2-x^2} \sqrt {-1+x^2}} \, dx\) [243]

Optimal. Leaf size=12 \[ -F\left (\left .\cos ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |2\right ) \]

[Out]

-(x^2)^(1/2)/x*EllipticF(1/2*(-2*x^2+4)^(1/2),2^(1/2))

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Rubi [A]
time = 0.00, antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {431} \begin {gather*} -F\left (\left .\text {ArcCos}\left (\frac {x}{\sqrt {2}}\right )\right |2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[2 - x^2]*Sqrt[-1 + x^2]),x]

[Out]

-EllipticF[ArcCos[x/Sqrt[2]], 2]

Rule 431

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[(-(Sqrt[c]*Rt[-d/c, 2]*Sqrt[a -
b*(c/d)])^(-1))*EllipticF[ArcCos[Rt[-d/c, 2]*x], b*(c/(b*c - a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c
] && GtQ[c, 0] && GtQ[a - b*(c/d), 0]

Rubi steps

\begin {align*} \int \frac {1}{\sqrt {2-x^2} \sqrt {-1+x^2}} \, dx &=-F\left (\left .\cos ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |2\right )\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(47\) vs. \(2(12)=24\).
time = 10.04, size = 47, normalized size = 3.92 \begin {gather*} \frac {\sqrt {1-x^2} \sqrt {1-\frac {x^2}{2}} F\left (\sin ^{-1}(x)|\frac {1}{2}\right )}{\sqrt {-2+3 x^2-x^4}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[2 - x^2]*Sqrt[-1 + x^2]),x]

[Out]

(Sqrt[1 - x^2]*Sqrt[1 - x^2/2]*EllipticF[ArcSin[x], 1/2])/Sqrt[-2 + 3*x^2 - x^4]

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Maple [A]
time = 0.10, size = 28, normalized size = 2.33

method	result	size
default	\(\frac {\EllipticF \left (\frac {x \sqrt {2}}{2}, \sqrt {2}\right ) \sqrt {-x^{2}+1}}{\sqrt {x^{2}-1}}\)	\(28\)
elliptic	\(\frac {\sqrt {-\left (x^{2}-1\right ) \left (x^{2}-2\right )}\, \sqrt {2}\, \sqrt {-2 x^{2}+4}\, \sqrt {-x^{2}+1}\, \EllipticF \left (\frac {x \sqrt {2}}{2}, \sqrt {2}\right )}{2 \sqrt {-x^{2}+2}\, \sqrt {x^{2}-1}\, \sqrt {-x^{4}+3 x^{2}-2}}\)	\(78\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(-x^2+2)^(1/2)/(x^2-1)^(1/2),x,method=_RETURNVERBOSE)

[Out]

EllipticF(1/2*x*2^(1/2),2^(1/2))*(-x^2+1)^(1/2)/(x^2-1)^(1/2)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-x^2+2)^(1/2)/(x^2-1)^(1/2),x, algorithm="maxima")

[Out]

integrate(1/(sqrt(x^2 - 1)*sqrt(-x^2 + 2)), x)

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-x^2+2)^(1/2)/(x^2-1)^(1/2),x, algorithm="fricas")

[Out]

0

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {\left (x - 1\right ) \left (x + 1\right )} \sqrt {2 - x^{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-x**2+2)**(1/2)/(x**2-1)**(1/2),x)

[Out]

Integral(1/(sqrt((x - 1)*(x + 1))*sqrt(2 - x**2)), x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-x^2+2)^(1/2)/(x^2-1)^(1/2),x, algorithm="giac")

[Out]

integrate(1/(sqrt(x^2 - 1)*sqrt(-x^2 + 2)), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.08 \begin {gather*} \int \frac {1}{\sqrt {x^2-1}\,\sqrt {2-x^2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((x^2 - 1)^(1/2)*(2 - x^2)^(1/2)),x)

[Out]

int(1/((x^2 - 1)^(1/2)*(2 - x^2)^(1/2)), x)

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